Cardioidal shaped brake band



Oct. 3 0, 1956 J. W. HOLDEMAN ET AL CARDIOIDAL SHAPED BRAKE BAND Filed Jan. 5, 1951 2 Sheets-Sheet l emefj' rr'eZ Oct. 30,l 1956 J. w. HOLDEMAN ET AL I 2,768,714

CARDIOIDAL SHAPED BRAKE BAND .2 Sheets-Sheet 2 Filed Jan. 5, 1951 inw Hol'cenzlz, Farrell and United States Patent O CARDIOIDAL SHAPED BRAKE BAND John W. Holdeman, Detroit, and Eugene F. Farrell,

Grosse Pointe, Mich., assignors to Borg-Warner Corporation, Chicago, Ill., a corporation of Illinois Application January 3, 1951, Serial No. 204,193

Claims. (Cl. 18S-259) The present invention relates to brake bands and more particularly to a brake band for braking a substantially circular drum element of an automotive vehicle transmission brake.

The main object of the present invention is to provide a brake band, which when constricted to its working shape by the application of force to each end thereof, assumes the shape of a substantially true circle so that substantially every increment of length of the band engages the drum.

Another object of the present invention is to provide a method of making a brake band which, when constricted to its working position, assumes substantially the shape of a true circle.

The practice generally followed heretofore in making brake bands for braking the rotatable drum elements of friction brakes has been to make the band substantially circular. The rst step generally followed has been to place a strip of flexible material with its ends joined together around an expansible mandrel. The mandrel, which is substantially circular, is then expanded in order to impart its circular shape to the band. When the band is subsequently cut, it expands and assumes a substantially circular shape. When forces are thereafter applied to the ends of the expanded band, in order to contract .it to its working shape, the band is no longer circular. Instead, the band becomes distorted due to the ben-ding moment resulting from the forces which must be applied to the ends of the band in order to bring them to their working position. This bending moment is greatest at the mid-point of the band and tapers off to zero at the ends. Thus more bending occurs at the mid-point of the band and this prevents its taking a circular shape in its Working position. When the brake lining is applied to the inner surface of the band the distortion mentioned above is great enough to produce a considerable variation `in brake lining thickness whenthe lining is subsequently bored to provide a circular inner periphery when the band is constricted to its Working position. Because of the variation in the thickness of the brake lining its life is shortened considerably. Since the brake band disclosed by the present invention is substantially circular when it is contracted to its Working position, the necessity of boring the brake lining after .it is cemented to the band is eliminated and because the thickness of the lining is substantially constant throughout its periphery, the life of the lining is greatly increased.

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mandrel, the periphery of which has the proper amount of reverse distortion to properly shape the brake band when the mandrel is expanded. The brake band, after having been shaped by the expanded mandrel, is shotpeened on its inner diameter and friction material is then cemented thereto. The bracket is then severed and the band is permitted to assume .its free shape. The shape of the free band is substantially the same as the shape of the expanded mandrel, however, the diameter is somewhat larger. When the band is thereafter compressed to its working shape by forces applied to the brackets on either end thereof it assumes substantially the shape of a true `circle so that boring of the lining is unnecessary.

The method disclosed by the present invention produces an economical, cold worked high strength band wherein camber between the band and the friction lining is practically eliminated. When the band is expanded by the expansible mandrel, the strength of the Weld between the bracket and the ends of the band is effectively tested. The present method allows considerably less accuracy in the initial forming of the band before the bracket is fastened in place. Inasmuch as the apparatus necessary for producing bands in accordance with the present invention is relatively inexpensive and because expensive machining operations to control the size of the band are unnecessary, the present invention provides a brake band that .is cheap to manufacture and has a long life.

The above and numerous other objects and advantages of the present invention will become apparent from the following detailed description when read in conjunction with the accompanying drawings, wherein:

Fig. l is a plan view of the expansible mandrel constructed in accordance with the principles of the present invention;

Fig. 2 is a graphic diagram showing the location of the centers of the different sections of the mandrel shown in Fig. l;

Fig. 3 is a side elevation view, shown partly in section, of the mandrel disclosed in Fig. l;

Fig. 4 shows an unlined band with its attached bracket forming a closed ring that has been stretched over an exence numerals designate identicalv parts in the several views, and referring in particular to Figs. l and 3, a non-circular expansible mandrel is indicated generally by reference numeral 10. The mandrel 10 comprises three individual segments 11, 12 and 13 and each of these segments covers an arc of approximately 120. EachV of the segments 11, 12 and 13 has a shoulder 14 for seating a brake band 15 around the mandrel. The inner periphery of `each of the segments 11, 12 and 13 is cone shaped n forreceiving a complimentary cone shaped'rexpanding The method disclosed by the present invention for making ,a brake band wherein the distortion of the band from a substantially circular shape, when it assumes itsworking position, is eliminated consists of preforming the band so as to give it the correct amount of reverse distortion when in its free position to therebyV cause it to assume member 16. As is obvious, when the member 16 is forced downwardly, as viewed Fig. 3, the segments 11, 12 and 13 are forced outwardly vto thereby engage and expand the band 15 until it` conforms substantially to the shape of the periphery offthe mandrel 10... s

As is apparent to those skilledin the art, a circularly shaped band does not contract to a true circle when forces are applied to its ends in order to bring themtogether. Because of the fact that the bending moment is greatest at the mid-point of the band and tapers off to yzero at the ends, more bending occurs at tbe mid-point of the band and prevents its taking a circular shape when its ends are forced towards one another. in order to provide a band having a free shape that will constrict to a circular shape when forces are applied to its ends, it is necessary.y to determine thefree shape of the band that will produce atrue circular shape when its ends are moved. towards one another. The mathematical calculation utilized in determining the free shape of a band that will compress to a circular shape when its ends are forced together will now be described.A

Referring to Fig. 8 a band or beam is shown in its compressed position in which its shape is a true` circle. Fig. 9 shows the band in its expanded position. When the band occupies its compressed position its center is O and lit has a constant radius r. In the free position of the'band its center is still at point O but its radius is now r-{-u, where u isV a variable. F is the force that must be .applied to eachrend of the band to bring the band to'its compressed position, and therefore the bending moment atany point on the periphery of the band isFx. However, from Fig. 8 it is apparent that x:r(1+cos 0) and therefore the bending moment MY is represented by the equation The dierential equation of a curved beam where E is,the modulus of elasticity and I is the moment of inertia of a section of the beam, is

The solution of this equation, using the rules for solving linear differential equations, is

Theconstants A and B depend only upon where the pole or center O is taken relative to the compressed ring. Since the line OA is an axis of symmetry, u has the u=l 1+so sin @HA cos @+B sin e Therefore, since u has the same value for :0 and 0:1r, A:0. When the band occupies its compressed position it assumes the shape of a true circle and under this condition u:0 throughout the peripheryV of the bandand therefore But differentiatingvEquation 3 above with respect to 0, it is apparent that F 3 -1;=ETl sin 0-l-0 cos 0) -A sin @+B eos 0=0 In this equation when 0 equals 0,

du -B and consequently 13:0. Therefore the solution for Equation 3 above is I -But since F, r, E and I are constants thisequation may be written `as follows u=a(1l0 sin 0) (4a) This equation therefore represents the shape'of the band when it assumes its relaxed position. the fact that the band assumes substantially the same Because of l shape as the expanded mandrel, even though the expanded band or free band has a somewhat greater radius, this equation also represents the shape of the expanded mandrel that is necessary to properly shape the band7 so that when it is compressed it will assume the shape of a true circle.

By assigning appropriate values te F, r, E and I the shape of the mandrel may be computed. The locus corresponding to the shape of the mandrel may then be plotted on a scale several times the size of the mandrel in order to produce the required degree of accuracy and then the locus may be approximated by circular arcs. When the locus is approximated by circular arcs it is apparent that the machining of tl e mandrel is greatly facilitated. It has been found, where the mandrel has a diameter of approximately 6 inches, that three substantially equal arcs are generally satisfactory for an angular expanse of zero to 180 and because of the fact that the curve is symmetrical, the range from 180 to 360 can be obtained by inversion. This method of graphically constructing the shape of the mandrel is somewhat difficult to carry out to the required degree of accuracy and therefore it is preferable to use the following analytical method for arriving at the shape of the mandrel.

The analytical method for determining the shape of the mandrel will now be described. First the x, y coordinates of the points on the curve formed by the periphery of the mandrel, for values of 0 equal to 0, 30, 60, 90, 120, and 180 degrees are computed. These x, y coordinates may be determined by merely solving Equation4 or 4a for the value of u corresponding to the different values of 0, and then the x coordinate may be determined from the equation x: (wl-u) cos 0 andrthe y coordinate may be determined from the equation y:(r-{-u) sin 0 Thus, the x, y coordinates for each 30 degree point 0n the periphery of the mandrel are known. The next step is to ydetermine the center of each circular are of 60 degrees from 0:0 to 180 degrees. Three points on each of these arcs have already been computed vand from the principles of analytic geometry the equations for a circle through three points having x, y coordinates equal respectively to (x0, y0), (x1, y1) and (x2, y2), and having a center Whose coordinates are (I1, k), are

k more2-Agni) where A1=x12xo2 B1=y12y02 C1=x22-x12; C2=x2x1 u D1`Zy22-y125 D2=32*"y1 From these equations the coordinates (h, k) representing the center of thev are through the points corresponding to 0:0, 30 and 60 degrees, may be computed. This same procedure may be used for determining the centers ofthe arcs through the pointscorresponding to 0:60, 90 and 120 degrees, and through the points corresponding` to 0:120, 150 and 180 degrees. The other half of the periphery of the mandrel is symmetrical egresjitt and of course may be readily computed. Having determined the coordinates for the center (h, k) for each arc, and since the coordinates (h, k) are known for one point on each arc, the radius of each arc may be determined from the following equation The mandrel 10, shown in Fig. 1, as has already been described, has three segments 11, 12 and 13. Each of the segments is divided into two sections of approximately 60 degrees each, the peripheries of each section having centers corresponding respectively to points 1- 6, shown in Fig. 2. The mean radius of the mandrel (shown in Fig. 1) is 3.173 inches and each 30 degree point on the periphery thereof has been computed in accordance with the analytical method described above Each 60 degree section of the mandrel 10 is identified in Fig. 1 by numerals 1-6 and the centers of curvature of each of these sections are identified by the corresponding ntunerals 1-6 in Fig. 2. In each of the Figs. l and 2 the positive x-:axis extends from the center of the figure to the right along the horizontal and the positive y-axis extends from the center of the figure upwardly along the vertical. The coordinates (h, k) representing the centers of the different sections and the radii for each of the sections, represented in inches, are given in the following table.

From the above table it is apparent that the periphery of the mandrel 10 is non-circular. However each of the sections 1-6 has a circular curvature and this facilitates the machining of the mandrel. Actually the shape of the periphery of the expanded mandrel 10 approximates very closely, Equation 4 or 4a set forth above and for all practical purposes a brake band formed by the mandrel shown in Fig. l has such a configuration that it `will assume the shape of substantially a true circle when its ends are forced together to bring it into engagement with a circular brake drum. It is apparent that the configuration of the mandrel 10 and of the brake band is substantially cardioidal. A In practicing the method of the present invention the mandrel 10 is first contracted. The brake band 15, having its ends V17 and 18 joined together by means of a bracket 19 secured, such as by welding, to each of the ends 17 and 18, and having a generally cylindrical configuration is placed in position over the mandrel 10. The expanding member 16 isthen driven downwardly to cause the segments 11, 12 and 13 to expand and engage the band 15. The cardioidal shape of the mandrel 10 is thereby imparted to the brake band 15. 'Ihe next method step, which is illustrated in Fig. 5, comprises `the fastening of a brake liner 20 onthe inner periphery of ltheband 15. The brake liner 20 consists of any suitable friction inducing material and may be fastened to the band by cement or Iany other suitable means.

c The next method step consists of cutting the bracket `19yso that it forms a pair of force receiving lugs `19a and 19h. When the bracket 19 is cut, the band 15 and liner 10 expand to the position asv shown 4'in Fig. V6. @The expanded or free band 15 retains its cardioidal shape even though its mean radius is somewhat greater than the mean radius of the expanded mandrel 10. When forces F are directed in a straight line towards each other against both of the force receiving lugs 19a and 19h the brake band 15, together with the liner 20, is compressed and assumes a working shape that is substantially circular. Because of this circular working shape of the band and liner it is unnecessary to bore the liner 20 in order that it will engage a circular brake drum at all points on the periphery of the liner. The reason that the band 15 assumes a substantially circular shape when compressed by the forces F is that mid-point 21 has the greatest radius of curvature and the radius of curvature progressively decreases towards either end 17 and `1S of the band. Since the bending moment at any point on the periphery of the band is equal to the force F multiplied by the distance between that point and the line of application of the force F, it is apparent that the bending moment is greatest towards the mid-point 21 of the band and progressively decreases towards the ends 17 and 18 of the band. Due to the greater radius or curvature of the middle portions of the band these portions must be bent more in order for them to approach a true circle and consequently since the bending moment is greater progressively from the end of the band to the midpoint thereof, the sections of the Vband havingthe greater radius of curvature are bent more, and thus the entire band is compressed to a substantially circular working position.

In the description above, of the method of making the brake band in accordance with the present invention, the liner 20 has been described as being secured to the band after the band itself has been expanded and shaped by the mandrel. It is to be noted, however, that the liner may just as well be attached to the band before the band shaping operation, if the liner is made of a material which will expand properly and will not be distorted during the shaping operation. This method of practicing the invention is particularly applicable to bi-metal bands when the two metals are more readily bonded together prior to the shaping operation and one of the metals serves as the liner.

The mandrel 10 is disclosed in the drawings and described above as comprising three substantially equiangular sections, however this is merely illustrative and we do not intend to be limited to this construction. We therefore desire to point out that it is within the realm of the present invention to utilize a mandrel or arbor consisting of any number of sections, each of which may be either substantially equiangular or some or all of which may be of substantially dierent sizes.

When the arbor is expanded to shape the band the tendency is for the portions of the periphery of the band which lie over the spaces between adjacent sections to be straight but when these spaces are narrow this tendency is negligible. The size of the spaces between adjacent sections, when the arbor is expanded, is inversely proportional to the number of sections in the arbor and it is therefore sometimes desirable to increase the number of sections to some number greater than three in order to decrease the size of the spaces. Even though the arbor iscomprised of more than three sections it has been found, if the overall shape of the arbor is the same as the overall shape'of the three section arbor described herein, that the arbor will produce an entirely satisfactory band. If it is desired to have the shape ofthe expanded mandrel more nearly approach Equation 4 or 4a it is within the realm of the present invention to have the periphery of the arbor conform to any number of cylindrical segments. It isV thus apparent that the arbor can be formed by first grinding the peripheral surface thereof to conform to the desired number of cylindri- .cal segments and then the arbor can be cut into the desired number of sections which may be equiangular or of different sizes.

As is apparent from the above description, the present invention provides a brake band which when moved to its working position assumes substantially the shape of a true circle, Therefore, the4 brake band engages the brakedrurn at allpoints on its periphery, and, because of: the uniform initial contact, the torque handling capacity of this bandis higher than the capacity of a band which is-V substantially circular in the free position. This is` true because the additional forceA required to make the conventional-band conform to the` drum, detracts from the force available for braking.

Itis contemplated that numerous changes may be made in the present invention without departing from the spirit or scope thereof.

We claim:

1. A brake band adapted to bel contracted into engagement with the outer periphery ofra rotatable drum for frictionally braking the drum and comprising continuous metal band around the periphery of the drum, said band having two ends and being expansible to cause disengagement of the bandfrom the drum and further being non-circular when in its expanded4 condition, and said band being contractible to a substantially circular shape to thereby engage the drum substantially simultaneously at all points on the band periphery when forces are applied to the ends of the band.

2. A brake band for substantially encircling a rotatable drum and adapted to be contracted into engagement therewith for braking rotation thereof and comprising acontinuous flexible metal band contractible to a substantially circular shape to engage and brake the drum and expansible to a relaxed shape that substantially approximates the equation where u is the change in the radius of the band, F is the force applied in order to contract the band to said circular shape, r is the mean radiusof the relaxed band, E is the modulus of elasticity of the band, I is the moment of inertia of the band, and 9Y is the angle subtended by a section of the band measured from the point on the band having the greatest radius of curvature.

3. A brake band contractible into engagement with a rotatable drum for braking the drum and comprising a continuous flexible band consisting of a strip of ilexible material for substantially encircling the drum and havin-g a pair of ends capable of receiving forces for contracting the band, said band, when in its relaxed condition with said forces not applied, comprising a plurality of sections having different radii of curvature such that the applicationof said forces to the ends of said band will constrict the band to a substantially circular shape to cause engagement of said band with sa-id drum substantially simultaneously at substantially all points on the periphery of 'said band.

4. A brake band contractible into engagement with a rotatable drum in order to brake the drum and comprising a continuous ilexible band having `a generally cardioidal relaxed shape, said band having a pair of ends capable of receiving forces for contracting the band, said band contracting upon the application of said forces to the ends thereof to a substantially circular shape, whereby said band enga-ges the drum at substantially all points on the periphery of said band.

5. A brake band adapted to be constricted into engagement with a rotatable drum in order to brake the drum and comprising a continuous flexible band having a pair of ends for receiving forces to cause constriction of said band until it engages the drum, said band,'when in its relaxed condition with said forces not applied, comprising Ia lplurality of sections of approximately Ithe same length on each side of a longitudinal mid-point of said band, the radii of curvature of said sections progressively decreasing from said longitudinal mid-,point to said ends of said band; whereby thel application of said forces to the ends of said bandcauses constriction thereof to a substantially circular shape sothat the band engages Ithe, drumat substantially all points on the periphery of the band.

6. A brake band for 'braking a substantially circular Irotatable drum comprising a strip of flexible material having a force receiving -lug on each end thereof, said strip comprising a plurality of substantially equal lengthy substantially circular sections, the radii of curvature of said sections increasing progressively from the ends of said strip toward thel mid-point of said strip, whereby application of a force lto each of said lugs compresses the band to a substantially circular shape.

7. A brake band for braking a substantially circular rotatable drumcomprising a strip of flexible material having a force receiving projection on each end thereof, said strip comprising a plurality of substantially cylindrically shaped segments, the radii of curvature of said segments progressively decreasing from the substantial mid-point of said strip toward each end thereof, whereby application offa force to each of the projections constricts the band to a substantially cylindrical shape.

8. A brake band for braking a rotatable drum comprisinga metal band adapted to be disposed around the periphery of the drum, said band having two ends and being expansible to cause disengagement of the band from the drum, said band, when in its expanded condition, having a reverse distortion such that forces applied to said ends to contract the band cause all points on the inner periphery thereof to simultaneously yengage the drum whereby all further forces applied to said ends are available to brake rotation of the drum.

9. A brake band for braking a rotatable drum comprising a flexible ba-nd consisting of a strip of flexible material :for substantially encircling the drum and-having a pair of 4ends capable of receiving forces for contracting, the band, said band, when in its relaxed condition with said forces not applied, comprising a plurality of sections havingdifferent radii of curvature such that the application of said forces to the ends of said band will constrict the band to a substantially circular shape to cause engagement of said band with said drum at substantially all points on the .periphery of said band whereby substantially all further forces applied to said ends are available to brake rotation of the drum.

10. A brake band for braking a rotatable drum comprising a flexible band having a generally cardioidal relaxed shape, said 'band having a pair of ends capable of receiving forces for contracting the band into engagement with said drum, said band contracting upon the application of initial forces to the ends thereof to a substantially circular shape, whereby said band engages the dr-urn at substantially all points on the periphery of said band upon the application of said initial forces and all further forces appliedvto said ends :are available to brake rotationv of the drum` References Cited in the le of this patent UNITED STATES PATENTS 1,529,087 Reynolds Mar. 10, 1925 1,654,736 Kister Jan. 3, 1928 1,659,908 Dean Feb. 21, 1928 1,732,630 Bennet Oct. 22, 1929 1,805,501@ Tatter May 19, 1931 1,926,064 Sawte'lle Sept. l2, `1933 2,046,306 lFykse June 30, 1936 2,289,311 Wellman July 7, 1942 2,382,947' Brozek Aug. 14, 1945 2,574,034 Heimann Nov. 6, 1951 2,705,059| Wilkinson Mar. 29, 1955 

